Question

A man standing in front of a mountain beats a drum at regular intervals. The drumming rate is gradually increased and he finds that the echo is not heard distinctly when the rate becomes $40/min$. He then moves near to the mountain by $90m$ and finds that the echo is again not heard when the drumming rate becomes $60/min$. If the distance (in meter) between the mountain and the initial position of the man is $x$. Find $\dfrac{x}{10}$

Answer 1

Here we need to calculate the distance between the man and the cliff to find $\dfrac{x}{10}$. We know from the definition of depend that it is the rate of change of distance with respect to time. Also since the sound wave from the source gets reflected from the mountains, it will travel two times the distance between the source and the mountain. Here, we will get two equations in terms of distance and speed. Hence using that we can solve the sum.

Formula used:
$speed=\dfrac{distance}{time}$

Complete step-by-step answer:
Let us assume that the man is standing at a distance $x$ from the mountain. Let us also assume that the sound waves from the beats travel at a speed $v$.
Then the distance travelled by the echo is $2x$ since the sound wave from the man travels to the mountain and returns back. If it takes $t$ for the echo to reach the man, then $t=\dfrac{2x}{v}$
Clearly, the sound is not heard due to the overlap of the sound wave and the reflected sound wave.
The sound is not heard when the rate is $40/min$. Then the interval between the successive beats is given as $\dfrac{60}{40}=1.5$
Then we can say that $\dfrac{2x}{v}=1.5$
$\implies 2x-1.5v=0$
$\implies 2x=1.5v$
Given that the man moves $90m$ towards the mountain, then the distance between the mountain and the man becomes $x- 90$
Then the distance travelled by the echo is given as $2(x-90)=2x-180$
Similarly, the sound is not heard when the rate is $60/min$. Then the interval between the successive beats is given as $\dfrac{60}{60}=1$
Then, we can say that $\dfrac{2x-180}{v}=1$
$\implies 2x-v=180$
Replacing the value of $2x$, we get, $1.5v-v=180$
$\implies 0.5v=180$
$\implies v=360m/s$
Then $2x=1.5\times 360$
$\implies x=270m$
Clearly, we get $x=270m$ and$v=360m/s$
Then $\dfrac{x}{10}=\dfrac{270}{10}=27m$
Hence the answer is $27m$.

Note: The rate of sound is nothing but the frequency of the sound. Since $f=\dfrac{1}{t}$, we can find time $t$. Here we are using the definition of the speed to calculate the speed of the wave. However note that, the distance travelled by the echo is always $2x$ with respect to the source.